## Tag `0CDC`

Chapter 51: Semistable Reduction > Section 51.14: Semistable reduction

Example 51.14.1. Let $R$ be a discrete valuation ring with uniformizer $\pi$. Given $n \geq 0$, consider the ring map $$ R \longrightarrow A = R[x, y]/(xy - \pi^n) $$ Set $X = \mathop{\rm Spec}(A)$ and $S = \mathop{\rm Spec}(R)$. If $n = 0$, then $X \to S$ is smooth. For all $n$ the morphism $X \to S$ is at-worst-nodal of relative dimension $1$ as defined in Algebraic Curves, Section 49.20. If $n = 1$, then $X$ is regular, but if $n > 1$, then $X$ is not regular as $(x, y)$ no longer generate the maximal ideal $\mathfrak m = (\pi, x, y)$. To ameliorate the situation in case $n > 1$ we consider the blowup $b : X' \to X$ of $X$ in $\mathfrak m$. See Divisors, Section 30.29. By construction $X'$ is covered by three affine pieces corresponding to the blowup algebras $A[\frac{\mathfrak m}{\pi}]$, $A[\frac{\mathfrak m}{x}]$, and $A[\frac{\mathfrak m}{y}]$.

The algebra $A[\frac{\mathfrak m}{\pi}]$ has generators $x' = x/\pi$ and $y' = y/\pi$ and $x'y' = \pi^{n - 2}$. Thus this part of $X'$ is the spectrum of $R[x', y'](x'y' - \pi^{n - 2})$.

The algebra $A[\frac{\mathfrak m}{x}]$ has generators $x$, $u = \pi/x$ subject to the relation $xu - \pi$. Note that this ring contains $y/x = \pi^n/x^2 = u^2\pi^{n - 2}$. Thus this part of $X'$ is regular.

By symmetry the case of the algebra $A[\frac{\mathfrak m}{y}]$ is the same as the case of $A[\frac{\mathfrak m}{y}]$.

Thus we see that $X' \to S$ is at-worst-nodal of relative dimension $1$ and that $X'$ is regular, except for one point which has an affine open neighbourhood exactly as above but with $n$ replaced by $n - 2$. Using induction on $n$ we conclude that there is a sequence of blowing ups in closed points $$ X_{\lfloor n/2 \rfloor} \to \ldots \to X_1 \to X_0 = X $$ such that $X_{\lfloor n/2 \rfloor} \to S$ is at-worst-nodal of relative dimension $1$ and $X_{\lfloor n/2 \rfloor}$ is regular.

The code snippet corresponding to this tag is a part of the file `models.tex` and is located in lines 5838–5887 (see updates for more information).

```
\begin{example}
\label{example-blowup}
Let $R$ be a discrete valuation ring with uniformizer $\pi$.
Given $n \geq 0$, consider the ring map
$$
R \longrightarrow A = R[x, y]/(xy - \pi^n)
$$
Set $X = \Spec(A)$ and $S = \Spec(R)$.
If $n = 0$, then $X \to S$ is smooth.
For all $n$ the morphism $X \to S$ is at-worst-nodal
of relative dimension $1$ as defined in
Algebraic Curves, Section \ref{curves-section-families-nodal}.
If $n = 1$, then $X$ is regular, but if $n > 1$, then $X$ is not
regular as $(x, y)$ no longer generate the maximal ideal
$\mathfrak m = (\pi, x, y)$. To ameliorate the situation
in case $n > 1$ we
consider the blowup $b : X' \to X$ of $X$ in $\mathfrak m$.
See Divisors, Section \ref{divisors-section-blowing-up}.
By construction $X'$ is covered by three affine pieces
corresponding to the blowup algebras $A[\frac{\mathfrak m}{\pi}]$,
$A[\frac{\mathfrak m}{x}]$, and $A[\frac{\mathfrak m}{y}]$.
\medskip\noindent
The algebra $A[\frac{\mathfrak m}{\pi}]$ has generators
$x' = x/\pi$ and $y' = y/\pi$ and $x'y' = \pi^{n - 2}$.
Thus this part of $X'$ is the spectrum of $R[x', y'](x'y' - \pi^{n - 2})$.
\medskip\noindent
The algebra $A[\frac{\mathfrak m}{x}]$ has generators $x$,
$u = \pi/x$ subject to the relation $xu - \pi$. Note that this ring
contains $y/x = \pi^n/x^2 = u^2\pi^{n - 2}$. Thus this part of
$X'$ is regular.
\medskip\noindent
By symmetry the case of the algebra $A[\frac{\mathfrak m}{y}]$ is
the same as the case of $A[\frac{\mathfrak m}{y}]$.
\medskip\noindent
Thus we see that $X' \to S$ is at-worst-nodal of relative dimension $1$
and that $X'$ is regular, except for one point which has an
affine open neighbourhood exactly as above but with $n$ replaced by $n - 2$.
Using induction on $n$ we conclude that there is a sequence of
blowing ups in closed points
$$
X_{\lfloor n/2 \rfloor} \to \ldots \to X_1 \to X_0 = X
$$
such that $X_{\lfloor n/2 \rfloor} \to S$ is
at-worst-nodal of relative dimension $1$ and
$X_{\lfloor n/2 \rfloor}$ is regular.
\end{example}
```

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